Question: A circle has a circumference of $12\pi$. It has an arc of length $\dfrac{27}{5}\pi$. What is the central angle of the arc, in radians? ${12\pi}$ ${\dfrac{9}{10}\pi}$ $\color{#DF0030}{\dfrac{27}{5}\pi}$
Answer: The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{\theta}{2 \pi} = \dfrac{27}{5}\pi \div 12\pi$ $\dfrac{\theta}{2 \pi} = \dfrac{9}{20}$ $\theta = \dfrac{9}{20} \times 2 \pi$ $\theta = \dfrac{9}{10}\pi$ radians